Polynomial values in Fibonacci sequences

Abstract

The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(ℤ) for any polynomial g which is not equivalent to a Dickson polynomial.